Riemannian geometry of noncommutative surfaces

Transcription

1 JOURNAL OF MATHEMATICAL PHYSICS 49, Riemannian geometry of noncommutative surfaces M. Chaichian, 1,a A. Tureanu, 1,b R. B. Zhang, 2,c and Xiao Zhang 3,d 1 Department of Physica Sciences, University of Hesinki and Hesinki Institute of Physics, P.O. Box 64, Hesinki, Finand 2 Schoo of Mathematics and Statistics, University of Sydney, Sydney, New South Waes 2006, Austraia 3 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing , Peope s Repubic of China Received 19 September 2007; accepted 11 June 2008; pubished onine 16 Juy 2008 A Riemannian geometry of noncommutative n-dimensiona surfaces is deveoped as a first step toward the construction of a consistent noncommutative gravitationa theory. Historicay, as we, Riemannian geometry was recognized to be the underying structure of Einstein s theory of genera reativity and ed to further deveopments of the atter. The notions of metric and connections on such noncommutative surfaces are introduced, and it is shown that the connections are metric compatibe, giving rise to the corresponding Riemann curvature. The atter aso satisfies the noncommutative anaog of the first and second Bianchi identities. As exampes, noncommutative anaogs of the sphere, torus, and hyperbooid are studied in detai. The probem of covariance under appropriatey defined genera coordinate transformations is aso discussed and commented on as compared to other treatments American Institute of Physics. DOI: / I. INTRODUCTION In recent years there has been much progress in deveoping theories of noncommutative geometry and exporing their appications in physics. Many viewpoints were adopted and different mathematica approaches were foowed by different researchers. Connes theory 1 see aso Ref. 2 formuated within the framework of C -agebras is the most successfu, which incorporates cycic cohomoogy and K-theory and gives rise to noncommutative versions of index theorems. Theories generaizing aspects of agebraic geometry were aso deveoped see, e.g., Ref. 3, for a review and references. A notion of noncommutative schemes was formuated, which seems to provide a usefu framework for deveoping noncommutative agebraic geometry. A major advance in theoretica physics in recent years was the deformation quantization of Poisson manifods by Kontsevich see Ref. 4, for the fina form of this work. This sparked intensive activities investigating appications of noncommutative geometries to quantum theory. Seiberg and Witten 5 showed that the antisymmetric tensor fied arising from massess states of strings can be described by the noncommutativity of a space-time, x,x = i, constant matrix, 1.1 where the mutipication of the agebra of functions is governed by the Moya product a Eectronic mai: masud.chaichian@hesinki.fi. b Eectronic mai: anca.tureanu@hesinki.fi. c Eectronic mai: rzhang@maths.usyd.edu.au. d Eectronic mai: xzhang@amss.ac.cn /2008/49 7 /073511/26/$ , American Institute of Physics Downoaded 04 Nov 2008 to Redistribution subject to ASCE icense or copyright; see

2 Chaichian et a. J. Math. Phys. 49, f g x = f x exp i 2 g x. 1.2 A considerabe amount of research was done both prior and after Ref. 5, and we refer to Refs. 6 and 7 for reviews and references. An earier and independent work is a semina paper by Dopicher et a., 8 which aid down the fundamentas of quantum fied theory on noncommutative space-time. These authors started with a theoretica examination of the ong hed beief by the physics community that the usua notion of space-time needed to be modified at the Panck scae and convincingy demonstrated that spacetime becomes noncommutative in that the coordinates describing space-time points become operators simiar to those in quantum mechanics. Therefore, noncommutative geometry is indeed a way to describe physics at Panck scae. A consistent formuation of a noncommutative version of genera reativity coud give insight into a gravitationa theory compatibe with quantum mechanics. A unification of genera reativity with quantum mechanics has ong been sought after but remains as eusive as ever despite the extraordinary progress in string theory for the ast two decades. The noncommutative geometrica approach to gravity coud provide an aternative route. Much work has aready been done in this genera direction, see, e.g., Refs and references therein. In particuar, different forms of noncommutative Riemannian geometries were proposed, 9,10,14,15 which retain some of the famiiar geometric notions such as metric and curvature. Noncommutative anaogs of the Hibert Einstein action were aso suggested 9,10,12,13 by treating noncommutative gravity as gauge theories. The noncommutative space-time with the Heisenberg-ike commutation reation 1.1 vioates Lorentz symmetry but was show 16,17 to have a quantum symmetry under the twisted Poincaré agebra. The Abeian twist eement F = exp i was used in Refs. 16 and 17 to twist the universa enveoping agebra of the Poincaré agebra, obtaining a noncommutative mutipication for the agebra of functions on the Poincaré group cosey reated to the Moya product 1.2. It is then natura to try to extend the procedure to other symmetries of noncommutative fied theory and investigate whether the concept of twist provides a new symmetry principe for noncommutative space-time. The same Abeian twist eement 1.3 was used in Refs. 9 and 10 for deforming the agebra of diffeomorphisms when attempting to obtain genera coordinate transformations on the noncommutative space-time. It is interesting that Refs. 9 and 10 proposed gravitationa theories which are different from the ow energy imit of strings. 18 However, based on physica arguments one woud expect the Moya product to be frame dependent and transform under the genera coordinate transformation. If the twist eement is chosen as 1.3, the Moya -product is fixed once for a. This is ikey to ead to probems simiar to those observed in Ref. 19 when one attempted to deform the interna gauge transformations with the same twist eement 1.3. Nevertheess twisting is expected to be a productive approach to the formuation of a noncommutative gravitationa theory when impemented consistenty. A covariant twist was proposed for interna gauge transformations in Ref. 20, but it turned out that the corresponding star product woud not be associative. Works on the noncommutative geometrica approach to gravity may be broady divided into two types. One type attempts to deveop noncommutative versions of Riemannian geometry axiomaticay that is, formay, whie the other adapts genera reativity to the noncommutative setting in an intuitive way. The probem is the ack of any safeguard against mathematica inconsistency in the atter type of works, and the same probem persists in the first type of works as we, since it is not cear whether nontrivia exampes exist which satisfy a the axioms of the formay defined theories. The aim of the present paper is to deveop a theory of noncommutative Riemannian geometry by extracting an axiomatic framework from highy nontrivia and transparenty consistent ex- Downoaded 04 Nov 2008 to Redistribution subject to ASCE icense or copyright; see

3 Riemannian geometry of noncommutative surfaces J. Math. Phys. 49, ampes. Our approach is mathematicay different from that of Refs. 9 and 10 and aso quite far removed from the quantum group theoretica noncommutative Riemannian geometry 15 see aso references in Ref. 15 and subsequent pubications by the same author. Reca that the two-dimensiona surfaces embedded in the Eucidean three-space provide the simpest yet nontrivia exampes of Riemannian geometry. The Eucidean metric of the three-space induces a natura metric for a surface through the embedding; the Levi Civita connection and the curvature of the tangent bunde of the surface can thus be described expicity for the theory of surfaces, see, e.g., the textbook Ref. 21. As a matter of fact, Riemannian geometry originated from Gauss work on surfaces embedded in three-dimensiona Eucidean space. More generay, Whitney s theorem enabes the embedding of any smooth Riemannian manifod as a high dimensiona surface in a fat Eucidean space of high enough dimension see, e.g., Theorems 9 and in Ref. 22. The embedding aso aows transparent construction and interpretation of a structures reated to the Riemannian metric of M as in the two-dimensiona case. This paper deveops noncommutative deformations of Riemannian geometry in the ight of Whitney s theorem. The first step is to deform the agebra of functions on a domain of the Eucidean space. We begin Sec. II by introducing the Moya agebra A, which is a noncommutative deformation 23 of the agebra of smooth functions on a region of R 2. The rest of Sec. II deveops a noncommutative Riemannian geometry for noncommutative anaogs of twodimensiona surfaces embedded in three-space. Working over the Moya agebra A, we show that much of the cassica differentia geometry for surfaces generaizes naturay to this noncommutative setting. In Sec. III, three iuminating exampes are constructed, which are, respectivey, noncommutative anaogs of the sphere, torus, and hyperbooid. Their noncommutative geometries are studied in detai. We emphasize that the embeddings pay a crucia roe in our current understanding of the geometry of the two-dimensiona noncommutative surfaces. The metric of a noncommutative surface is constructed in terms of the embedding; the necessity of a eft connection and aso a right connection then naturay arises; even the definition of the curvature tensor is forced upon us by the context. Indeed, the extra information obtained by considering embeddings provides the guiding principes, which are acking up to now, for buiding a theory of noncommutative Riemannian geometry. Once the noncommutative Riemannian geometry of the two-dimensiona surfaces is sorted out, its generaization to the noncommutative geometries corresponding to n-dimensiona surfaces embedded in spaces of higher dimensions is straightforward. This is discussed in Sec. VI. Reca that the basic principe of genera reativity is genera covariance. We study in Sec. V the genera coordinate transformations for noncommutative surfaces, which are brought about by gauge transformations on the underying noncommutative associative agebra A over which noncommutative geometry is constructed. A new feature here is that the genera coordinate transformations affect the mutipication of the underying associative agebra A as we, turning it into another agebra nontriviay isomorphic to A. We make comparison with cassica Riemannian geometry, showing that the gauge transformations shoud be considered as noncommutative anaogs of diffeomorphisms. The theory of surfaces deveoped over the deformation of the agebra of smooth functions on some region in R n now suggests a genera theory of noncommutative Riemannian geometry of n-dimensiona surfaces over arbitrary unita associative agebras with derivations. We present an outine of this genera theory in Sec. IV. We concude this section with a remark on the presentation of the paper. As indicated above, we start from the simpest nontrivia exampes of noncommutative Riemannian geometries and graduay extend the resuts to buid up a theory of generaity. This experimenta approach is not the optima format for presenting mathematics, as a specia cases repeat the same pattern. However, it has the distinctive advantage that the genera theory obtained in this way stands on firm ground. Downoaded 04 Nov 2008 to Redistribution subject to ASCE icense or copyright; see

4 Chaichian et a. J. Math. Phys. 49, II. NONCOMMUTATIVE SURFACES AND THEIR EMBEDDINGS The first step in constructing noncommutative deformations of Riemannian geometry is the deformation of agebras of functions. Let us fix a region U in R 2 and write the coordinate of a point t in U as t 1,t 2. Let h be a rea indeterminate, and denote by R h the ring of forma power series in h. Let A be the set of the forma power series in h with coefficients being rea smooth functions on U. Namey, every eement of A is of the form i 0 f i h i, where f i are smooth functions on U. Then A is an R h -modue in an obvious way. Given any two smooth functions f and g on U, we denote by fg the usua pointwise product of the two functions. We aso define their star product or more precisey, Moya product by f g = im t t exp h t 1 t 2 t 2 t 1 f t g t, 2.1 where the exponentia exp h / t 1 / t 2 / t 2 / t 1 is to be understood as a power series in the differentia operator / t 1 / t 2 / t 2 / t 1. More expicity, et be R-inear maps defined by p :A/h A A/h A A/h A, p = 0,1,2, p f,g = im t tp! t 1 t 2 p f t g t. t 2 t 1 Then f g= p=0 h p p f,g. It is evident that f g ies in A. We extend this star product R h ineary to a eements in A by etting f i h i g j h j := f i g j h i+j. It has been known since the eary days of quantum mechanics that the Moya product is associative see, e.g., Ref. 4, for a reference, thus we arrive at an associative agebra over R h, which is a deformation 23 of the agebra of smooth functions on U. We sha usuay denote this associative agebra by A, but when it is necessary to make expicit the mutipication of the agebra, we sha write it as A,. Remark 2.1: For the sake of being expicit, we restrict ourseves to consider the Moya product defined by 2.1 ony in this section. As we sha see in Secs. IV and V, the theory of noncommutative surfaces to be deveoped in this paper extends to more genera star products over agebras of smooth functions. Write i for / t i and extend R h -ineary the operators i to A. One can easiy verify that for smooth functions f and g, i f g = i f g + f i g, that is, the operators i are derivations of the agebra A. Let A 3 =A A A. There is a natura two-sided A-modue structure on A 3 R h A 3, defined for a a,b A, and X Y A 3 R h A 3 by a X Y b=a X Y b. Define the map 2.3 A 3 R h A 3 A, a,b,c f,g,h a f + b g + c h, 2.4 and denote it by. This is a map of two-sided A-modues in the sense that for any X,Y A 3 and a,b A, a X Y b =a X Y b. We sha refer to this map as the dot product. Let X= X 1,X 2,X 3 be an eement of A 3, where the superscripts of X 1, X 2, and X 3 are not powers but are indices used to abe the components of a vector as in the usua convention in differentia geometry. We set i X= i X 1, i X 2, i X 3 and define the foowing 2 2-matrix over A, Downoaded 04 Nov 2008 to Redistribution subject to ASCE icense or copyright; see

5 Riemannian geometry of noncommutative surfaces J. Math. Phys. 49, g = g 11 g 12 g 21 g 22, g ij = i X j X. 2.5 Let g 0 =g mod h, which is a 2 2-matrix of smooth functions on U. Definition 2.2: We ca an eement X A 3 the noncommutative embedding in A 3 of a noncommutative surface if g 0 is invertibe for a t U. In this case, we ca g the metric of the noncommutative surface. Given a noncommutative surface X with a metric g, there exists a unique 2 2-matrix g ij over A which is the right inverse of g, i.e., g ij g jk = i k, where we have used Einstein s convention of summing over repeated indices. To see the existence of the right inverse, we write g ij = p h p g ij p and g ij = p h p g ij p, where g ij 0 is the inverse of g ij 0. Now in terms of the maps k defined by 2.2, we have which is equivaent to i k = g ij g jk = q h q m+n+p=q p g ij m,g jk n, q g ij q = q n n=1 m=0 g ik 0 n g k m,g j q n m. Since the right-hand side invoves ony g j r with r q, this equation gives a recursive formua for the right inverse of g. In the same way, we can aso show that there aso exists a unique eft inverse of g. It foows from the associativity of mutipication of matrices over any associative agebra that the eft and right inverses of g are equa. Definition 2.3: Given a noncommutative surface X, et E i = i X, i = 1,2, and ca the eft A-modue TX and right A-modue T X defined by TX = a E 1 + b E 2 a,b A, T X = E 1 a + E 2 b a,b A the eft and right tangent bundes of the noncommutative surface, respectivey. Then TX R h T X is a two-sided A-modue. Proposition 2.4: The metric induces a homomorphism of two-sided A-modues, g:tx R h T X A, defined for any Z=z i E i TX and W=E i w i T X by Z W g Z,W = z i g ij w j. It is easy to see that the map is indeed a homomorphism of two-sided A-modues, and it ceary coincides with the restriction of the dot product to TX R h T X. Since the metric g is invertibe, we can define which beong to TX and T X, respectivey. Then E i = g ij E j, Ẽ i = E j g ji, 2.6 Downoaded 04 Nov 2008 to Redistribution subject to ASCE icense or copyright; see

6 Chaichian et a. J. Math. Phys. 49, g E i,e j = j i = g E j,ẽ i, g E i,ẽ j = g ij. Now any Y A 3 can be written as Y =y i E i +Y, with y i =Y Ẽ i and Y =Y y i E i. We sha ca y i E i the eft tangentia component and Y the eft norma component of Y. Let TX = N A 3 N E i =0, i, which is ceary a eft A-submodue of A 3. In a simiar way, we may aso decompose Y into Y =E i ỹ i +Ỹ with the right tangentia component of Y given by ỹ i =E i Y and the right norma component by Ỹ =Y E i ỹ i. Let T X = N A 3 E i N =0, i, which is a right A-submodue of A 3. Therefore, we have the foowing decompositions: A 3 = TX TX, as eft A modue, A 3 = T X T X, as right A modue. 2.7 It foows that the tangent bundes are finitey generated projective modues over A. Foowing the genera phiosophy of noncommutative geometry, 1 we may regard finitey generated projective modues over A as vector bundes on the noncommutative surface. This justifies the terminoogy of eft and right tangent bundes for TX and T X. In fact, TX and T X are free eft and right A-modues, respectivey, as E 1 and E 2 form A-bases for them. Consider TX, for exampe. If there exists a reation a i E i =0, where a i A, we have a i E i E j =a i g ij =0, j. The invertibiity of the metric then eads to a i =0, i. Since E 1 and E 2 generate TX, they indeed form an A-basis of TX. One can introduce connections to the tangent bundes by foowing the standard procedure in the theory of surfaces. 21 Definition 2.5: Define operators i :TX TX, i = 1,2, by requiring that i Z be equa to the eft tangentia component of i Z for a Z TX. Simiary define i:t X T X, i = 1,2, by requiring that iz be equa to the right tangentia component of i Z for a Z T X. Ca the set consisting of the operators i i a connection on TX T X. The foowing resut justifies the terminoogy. Lemma 2.6: For a Z TX, W T X and f A, i f Z = i f Z + f i Z, i W f = W i f + iw f. 2.8 Proof: By the Leibniz rue 2.3 for i, i f Z = i f Z + f i Z, i W f = W i f + W i f. The emma immediatey foows from the tangentia components of these reations under the decompositions 2.7. In order to describe the connections more expicity, we note that there exist ij and ij in A such that Downoaded 04 Nov 2008 to Redistribution subject to ASCE icense or copyright; see

7 Riemannian geometry of noncommutative surfaces J. Math. Phys. 49, i E j = k ij E k, ie j = E k k ij. 2.9 Because the metric is invertibe, the eements k ij and k ij are uniquey defined by Eq We have k ij = i E j E k, k ij = E k i E j It is evident that k ij and k ij are symmetric in the indices i and j. The foowing cosey reated objects wi aso be usefu ater: ijk = i E j E k, ijk = E k i E j. In contrast to the commutative case, k ij and k ij do not coincide, in genera. We have where k ij = c ij g k + ij g k, k ij = g k c ij g k ij, c ij = 1 2 ig j + j g i g ji, ij = 1 2 ie j E E i E j. We sha ca ij the noncommutative torsion of the noncommutative surface. Therefore the eft and right connections invove two parts. The part c ij depends on the metric ony, whie the noncommutative torsion embodies extra information. For a noncommutative surface embedded in A 3, the noncommutative torsion depends expicity on the embedding. In the cassica imit with h =0, k ij vanishes and both k ij and k ij reduce to the standard Levi Civita connection. We have the foowing resut. Proposition 2.7: The connections are metric compatibe in the foowing sense: This is equivaent to the fact that i g Z,Z = g i Z,Z + g Z, iz, Z TX,Z T X i g jk ijk ikj = Proof: Since g is a map of two-sided A-modues, it suffices to prove 2.11 by verifying the specia case with Z=E j and Z =E k. We have i g E j,e k = i E j E k = i E j E k + E j i E k = g i E j,e k + g E j, ie k, where the second equaity is equivaent This proves both statements of the proposition. Remark 2.8: In contrast to the commutative case, Eq by itsef is not sufficient to uniquey determine the connections ijk and ijk; the noncommutative torsion needs to be specified independenty. This is simiar to the situation in supergeometry, where torsion is determined by other considerations. At this point we shoud reate to the iterature. The metric introduced here resembes simiar notions in Refs. 14 and 24 26; aso our eft and right connections and their metric compatibiity have much simiarity with Definitions 2 and 3 in Ref. 24. However, there are crucia differences. Our eft right tangent bunde is a eft right A-modue ony, whie in Refs. 24 and 25 there is ony one tangent bunde T which is a bimodue over some agebra or Hopf agebra B. The metrics defined in Refs. 14 and are maps from T B T to B. Remark 2.9: A noteworthy feature of the metric in Ref. 14 is that a particuar moving frame can be chosen to make a the components of the metric centra see Eq in Ref. 14. Inthe context of the Moya agebra, this amounts to that the metric is a constant matrix. Downoaded 04 Nov 2008 to Redistribution subject to ASCE icense or copyright; see

8 Chaichian et a. J. Math. Phys. 49, A. Curvatures and second fundamenta form Let i, j := i j j i and i, j := i j j i. Straightforward cacuations show that for a f A, i, j f Z = f i, j Z, Z TX, i, j W f = i, j W f, W T X. Ceary the right-hand side of the first equation beongs to TX, whie that of the second equation beongs to T X. We restate these important facts as a proposition. Proposition 2.10: The foowing maps i, j :TX TX, i, j :T X T X are eft and right A -modue homomorphisms, respectivey. Since TX T X is generated by E 1 and E 2 as a eft right A-modue, by Proposition 2.10, we can aways write i, j E k = R kij E, i, j E k = E R kij 2.13 for some R kij,r kij A. Definition 2.11: We refer to R kij and R kij, respectivey, as the Riemann curvatures of the eft and right tangent bundes of the noncommutative surface X. The Riemann curvatures are uniquey determine by the reations In fact, we have R kij Simpe cacuations yied the foowing resut. Lemma 2.12: = g i, j E k,ẽ, R kij = g E, i, j E k R kij = j ik ik p jp + i jk + jk p ip, R kij = j ik jp ik p + i jk + ip p jk. Proposition 2.13 : Let R kij =R p kij g p and R kij= g kp R ij p. The Riemann curvatures of the eft and right tangent bundes coincide in the sense that R kij =R kij. Proof: By Proposition 2.7, we have R kij = i j j i E k E, which can be rewritten as R kij = i j E k E j E k ie j i E k E + i E k je = i j E k E + E k je E k i je j i E k E + E k ie + E k j ie. Again by Proposition 2.7, the first term on the far right-hand side can be written as i j g k, and the third term can be written as i j g k. Thus they cance out, and we arrive at R kij = E k i j j i E = R kij. Because of the proposition, we ony need to study the Riemannian curvature on one of the tangent bundes. Note that R kij = R kji, but there is no simpe rue to reate R kij to R kij in contrast to the commutative case. Definition 2.14: Let R ij = R p ipj, R = g ji R ij, 2.15 Downoaded 04 Nov 2008 to Redistribution subject to ASCE icense or copyright; see

9 Riemannian geometry of noncommutative surfaces J. Math. Phys. 49, and ca them the Ricci curvature and scaar curvature of the noncommutative surface, respectivey. Then obviousy R ij = g j, E i,ẽ, R = g i, k E i,ẽ k In the theory of cassica surfaces, the second fundamenta form pays an important roe. A simiar notion exists for noncommutative surfaces. Definition 2.15: We define the eft and right second fundamenta forms of the noncommutative surface X by h ij = i E j k ij E k, It foows from Eq. 2.9 that h ij = i E j E k k ij h ij E k =0, E k h ij = Remark 2.16: Both the eft and right second fundamenta forms reduce to h 0 ij N in the commutative imit, where h 0 ij is the standard second fundamenta form and N is the unit norma vector. The Riemann curvature R kij = i j j i E k E can be expressed in terms of the second fundamenta forms. Note that By Definition 2.15, R kij = j E k i E j E k ie i E k j E + i E k je. R kij = j E k h i i E k h j = j E k + h jk h i i E k + h ik h j. Equation 2.18 immediatey eads to the foowing resut. Lemma 2.17: The foowing generaized Gauss equation hods: R kij = h jk h i h ik h j Before cosing this section, we mention that the Riemannian structure of a noncommutative surface is a deformation of the cassica Riemannian structure of a surface by incuding quantum corrections. The embedding into A 3 is not subject to any constraints as the genera theory stands. However, one may consider particuar noncommutative surfaces with embeddings satisfying extra symmetry requirements simiar to the way in which various star products on R 3 were obtained from the Moya product on R 4 in Secs. 4 and 5 in Ref. 27. III. EXAMPLES In this section, we consider in some detai three concrete exampes of noncommutative surfaces: the noncommutative sphere, torus, and hyperbooid. A. Noncommutative sphere Let U= 0, 0,2, and we write and for t 1 and t 2, respectivey. Let X, = X 1,,X 2,,X 3, be given by = sin cos sin sin cosh 2h cos X,,,, 3.1 cosh h cosh h cosh h with the components being smooth functions in, U. It can be shown that X satisfies the foowing reation: Downoaded 04 Nov 2008 to Redistribution subject to ASCE icense or copyright; see

10 Chaichian et a. J. Math. Phys. 49, X 1 X 1 + X 2 X 2 + X 3 X 3 = Thus we may regard the noncommutative surface defined by X as an anaog of the sphere S 2.We 2 sha denote it by S h and refer to it as a noncommutative sphere. We have = E 1 cos cos cos sin cosh 2h sin,,, cosh h cosh h cosh h E 2 = sin sin sin cos,,0 cosh h cosh h. 2 The components g ij =E i E j of the metric g on S h can now be cacuated, and we obtain g 11 =1, g 22 = sin 2 sinh2 h cosh 2 h cos2, sinh h g 12 = g 21 = cosh h sin2 cos 2. The components of this metric commute with one another as they depend on ony. Thus it makes sense to consider the usua determinant G of g. We have G = sin 2 + tanh 2 h cos 2 2 cos 2 = sin tanh 2 h 1 4cos 2. The inverse metric is given by g 11 sin 2 tanh 2 h cos 2 =, sin 2 + tanh 2 h cos 2 2 cos 2 g 22 1 =, sin 2 + tanh 2 h cos 2 2 cos 2 g 12 = g 21 tanh h cos 2 =. sin 2 + tanh 2 h cos 2 2 cos 2 Now we determine the connection and curvature tensor of the noncommutative sphere. The computations are quite engthy, thus we ony record the resuts here. For the Christoffe symbos, we have 111 = 111 =0, 112 = 112 = sin 2 tanh h, 121 = 121 = sin 2 tanh h, 122 = 122 = 1 2 sin tanh2 h, 211 = 211 = sin 2 tanh h, 212 = 212 = 1 2 sin tanh2 h, Downoaded 04 Nov 2008 to Redistribution subject to ASCE icense or copyright; see

11 Riemannian geometry of noncommutative surfaces J. Math. Phys. 49, = 221 = 1 2 sin tanh2 h, 222 = 222 = sin 2 tanh h. Note that cf. Remark 2.8. We now find the asymptotic expansions of the curvature tensors with respect to h, R 1112 =2h cos 2 h 3 + O h 4, R 2112 = sin cos 2 cos 4 h 2 + O h 4, R 1212 = sin cos 2 cos 4 h 2 + O h 4, R 2212 = 2sin 2 h cos 2 4 cos 4 h 3 + O h 4. We can aso compute asymptotic expansions of the Ricci curvature tensor, R 11 = cos 2 h 2 + O h 4, R 21 = 2 cos 2 h cos 2 6 cos 4 h 3 + O h 4, R 12 = 2 + cos 2 h cos cos 4 h 3 + O h 4, and the scaar curvature R 22 = sin cos 2 2 cos 4 h 2 + O h 4, R = cos 2 h 2 + O h 4. 2 By setting h =0, we obtain from the various curvatures of S h the corresponding objects for the usua sphere S 2. This is a usefu check that our computations above are accurate. B. Noncommutative torus This time we sha take U= 0,2 0,2 and denote a point in U by,. Let X, = X 1,,X 2,,X 3, be given by X, = a + sin cos, a + sin sin,cos, 3.3 where a 1 is a constant. Cassicay X is the torus. When we extend scaars from R to R h and impose the star product on the agebra of smooth functions, X gives rise to a noncommutative 2 torus, which wi be denoted by T h. We have E 1 = cos cos,cos sin, sin, E 2 = a + sin sin, a + sin cos,0. 2 The components g ij =E i E j of the metric g on T h take the form g 11 = 1 + sinh 2 h cos 2, g 22 = a + cosh h sin 2 sinh 2 h cos 2, Downoaded 04 Nov 2008 to Redistribution subject to ASCE icense or copyright; see

12 Chaichian et a. J. Math. Phys. 49, g 12 = g 21 = sinh h cosh h cos 2 + a sinh h sin. As they depend ony on, the components of the metric commute with one another. The inverse metric is given by g 11 = a + cosh h sin 2 sinh 2 h cos 2, G g 22 = 1 + sinh2 h cos 2, G g 12 = g 21 = sinh h cosh h cos 2 + a sinh h sin, G where G is the usua determinant of g given by G = sin + a cosh h 2 a 2 sin 2 sinh 2 h. Now we determine the curvature tensor of the noncommutative torus. The computations can be carried out in much the same way as in the case of the noncommutative sphere, and we merey record the resuts here. For the connection, we have 111 = sin 2 sinh 2 h, 112 = a cos sinh h + sin 2 sinh h cosh h, 121 = sin 2 sinh h cosh h, 122 = a cos cosh h sin 2 cosh 2h, 211 = sin 2 sinh h cosh h, 212 = a cos cosh h sin 2 cosh 2h, 221 = a cos cosh h 1 2 sin 2 cosh h, 222 =2a cos sinh h + sin 2 sinh h cosh h. We can find the asymptotic expansions of the curvature tensors with respect to h, 2 sin 1+a sin R 1112 = h + O h 3, a + sin R 2112 = sin a + sin + O h 2, R 1212 = sin a + sin + O h 2, R 2212 = 2sin 2 1+a sin h + O h 3. We can aso compute asymptotic expansions of the Ricci curvature tensor, R 11 = sin a + sin + O h 2, R 21 = sin 3a +5a cos 5+2a2 sin + sin 3 h 2 a + sin 2 + O h 3, Downoaded 04 Nov 2008 to Redistribution subject to ASCE icense or copyright; see

13 Riemannian geometry of noncommutative surfaces J. Math. Phys. 49, sin a + cos 2 + a sin R 12 = h + O h 3, a + sin and the scaar curvature, R 22 = sin a + sin + O h 2, 2 sin R = a + sin + O h 2. 2 By setting h =0, we obtain from the various curvatures of T h the corresponding objects for the usua torus T 2. C. Noncommutative hyperbooid Another simpe exampe is the noncommutative anaog of the hyperbooid described by X = x,y, 1+x 2 +y 2. One may aso change the parametrization and consider instead X r, = sinh r cos,sinh r sin,cosh r on U= 0, 0,2, where a point in U is denoted by r,. When we extend scaars from R to R h and impose the star product on the agebra of smooth functions defined by 2.1 with 2 t 1 =r and t 2 =, X gives rise to a noncommutative hyperbooid, which wi be denoted by H h.we have E 1 = cosh r cos,cosh r sin,sinh r, 3.4 E 2 = sinh r sin,sinh r cos,0. 2 The components g ij =E i E j of the metric g on H h take the form g 11 = cos 2 h cosh 2r, g 22 = cos 2h cosh 2r, g 12 = g 21 = 1 2 sin 2h cosh 2r. As they depend ony on r, the components of the metric commute with one another. The inverse metric is given by g 11 = sec2 h 1 2h 2 sinh 2 r cos cosh 2r, g 22 = 1 sinh 2 r, g 12 = g 21 tan h = sinh 2 r. Now we determine the curvature tensor of the noncommutative hyperbooid. For the connection, we have Downoaded 04 Nov 2008 to Redistribution subject to ASCE icense or copyright; see

14 Chaichian et a. J. Math. Phys. 49, = cos 2 h sinh 2r, 112 = 1 2 sin 2h sinh 2r, 121 = 1 2 sin 2h sinh 2r, 122 = 1 2 cos 2h sinh 2r, 211 = 1 2 sin 2h sinh 2r, 212 = 1 2 cos 2h sinh 2r, 221 = 1 2 cos 2h sinh 2r, 222 = 1 2 sin 2h sinh 2r. We can find the asymptotic expansions of the curvature tensors with respect to h, R 1112 = 2 cosh 2r h + O h 2, R 2112 = sinh2 r cosh 2r + O h 2, R 1212 = sinh2 r cosh 2r + O h 2, R 2212 = cosh 2r + sinh2 2r h + O h 3. cosh 2r We can aso compute asymptotic expansions of the Ricci curvature tensor, R 11 = 1 cosh 2r + O h 2, R 21 = coth2 r 2 cosh 2r 1 h + O h 2, cosh 2r R 12 = cosh 2r +2 cosh 2 2r h + O h 2, and the scaar curvature, R 22 = sinh2 r cosh 2 2r + O h 2, 2 R = cosh 2 2r + O h 2. 2 By setting h =0, we obtain from the various curvatures of H h the corresponding objects for the usua hyperbooid H 2. IV. NONCOMMUTATIVE n-dimensional SURFACES One can readiy generaize the theory of Sec. II to higher dimensions, and we sha do this here. Noncommutative Bianchi identities wi aso be obtained. Downoaded 04 Nov 2008 to Redistribution subject to ASCE icense or copyright; see

15 Riemannian geometry of noncommutative surfaces J. Math. Phys. 49, Again for the sake of expicitness we restrict attention to the Moya product on the smooth functions. However, as we sha see in Sec. V, it wi be necessary to consider more genera star products in order to discuss genera coordinate transformations of noncommutative surfaces. A. Noncommutative n-dimensiona surfaces We take a region U in R n for a fixed n and write the coordinate of t U as t 1,t 2,...,t n. Let A denote the set of the smooth functions on U taking vaues in R h. Fix any constant skew symmetric n n matrix. The Moya product on A is defined by the foowing generaization of Eq. 2.1 : f g = im t t exp h ij i f t g t j ij 4.1 for any f,g A. Such a mutipication is known to be associative. Since is a constant matrix, the Leibniz rue 2.3 remains vaid in the present case, i f g = i f g + f i g. For any fixed positive integer m, we can define a dot product, :A m R h A m A m 4.2 by generaizing 2.4 to A B = a i b i for a A= a 1,...,a m and B= b 1,...,b m in A m.as before, the dot product is a map of two-sided A-modues. Assume m n. For X A m,weete i = i X and define g ij =E i E j. Denote by g= g ij the n n matrix with entries g ij. Definition 4.1: If g mod h is invertibe over U, we sha ca X a noncommutative n-dimensiona surface embedded in A m and ca g the metric of X. The discussion on the metric in Sec. II carries over to the present situation; in particuar, the invertibiity of g mod h impies that there exists a unique inverse g ij. Now as in Sec. II, we define the eft tangent bunde TX right tangent bunde T X of the noncommutative surface as the eft right A-submodue of A m generated by the eements E i. The fact that the metric g beongs to GL n A enabes us to show that the eft and right tangent bundes are projective A-modues. The connection i on the eft tangent bunde wi be defined in the same way as in Sec. II, namey, by the composition of the derivative i with the projection of A m onto the eft tangent bunde. The connection i on the right tangent bunde is defined simiary. Then i and i satisfy the anaogous Eq. 2.8, and are compatibe with the metric in the same sense as Proposition 2.7. One can show that i, j :TX TX, i, j :T X T X are eft and right A-modue homomorphisms, respectivey. This aows us to define Riemann curvatures of the tangent bundes as in Eq Then the formuas given in Lemma 2.12 are sti vaid when the indices in the formuas are assumed to take vaues in 1,2,...,n. Furthermore, the eft and right Riemann curvatures remain equa in the sense of Proposition Remark 4.2: One may define a dot product :A m R h A m A m with a Minkowski signature by m 1 A B = a 0 b 0 a i b i i=1 for any A= a 0,a 1,...,a m 1 and B= b 0,b 1,...,b m 1 in A m. This is sti a map of two-sided A-modues. Then the aforedeveoped theory can be adapted to this setting, eading to a theory of noncommutative surfaces embedded in A m with a Minkowski signature. Downoaded 04 Nov 2008 to Redistribution subject to ASCE icense or copyright; see

16 Chaichian et a. J. Math. Phys. 49, For the sake of being concrete, we sha consider ony noncommutative surfaces with Eucidean signature hereafter. B. Bianchi identities We examine properties of the Riemann curvature for arbitrary n and m. The main resut in this subsection is the noncommutative anaogs of Bianchi identities. Define E i and Ẽ as in 2.6. Then p E = pk E k, These reations wi be needed presenty. Let pẽ = Ẽ k pk. 4.3 R kij;p = p R kij r pk R rij pi i Theorem 4.3: The Riemann curvature R jk R ijk + R jki + R kij r R rjk =0, R kij;p r pj R rki r + R kij satisfies the foowing reations: + R kjp;i rp R kpi;j =0, 4.5 which wi be referred to as the first and second noncommutative Bianchi identities, respectivey. Proof: It foows from the reation i E j = j E i that This immediatey eads to i, j E k + j, k E i + k, i E j =0. g i, j E k,ẽ + g j, k E i,ẽ + g k, i E j,ẽ =0. Using the definition of the Riemann curvature in this reation, we obtain the first Bianchi identity. To prove the second Bianchi identity, note that p R kij + g p i, j E k,ẽ + g i, j E k, pẽ =0. Cycic permutations of the indices p,i, j ead to two further reations. Adding a the three reations together, we arrive at p R kij + g i, j p E k,ẽ + g i, j E k, pẽ, i R kjp + g j, p i E k,ẽ + g j, p E k, iẽ, j R kpi + g p, i j E k,ẽ + g p, i E k, jẽ =0, 4.6 where we have used the foowing variant of the Jacobian identity: p i, j + i j, p + j p, i = i, j p + j, p i + p, i j. By a tedious cacuation one can show that Q ijkp := j, k p E i + k, i p E j + p, k i E j + k, j i E p + i, k j E p + k, p j E i is identicay zero. Now we add g Q ijkp,ẽ to the eft-hand side of 4.6, obtaining an identity with 15 terms on the eft. Then the second Bianchi identity can be read off this equation by recaing 4.3. Downoaded 04 Nov 2008 to Redistribution subject to ASCE icense or copyright; see

17 Riemannian geometry of noncommutative surfaces J. Math. Phys. 49, C. Einstein s equation Reca that in cassica Riemannian geometry, the second Bianchi identity suggests the correct form of Einstein s equation. Let us make some preiminary anaysis of this point here. As we ack guiding principes for constructing an anaog of Einstein s equation, the materia of this subsection is of a rather specuative nature. In Sec. II, we introduced the Ricci curvature R ij and scaar curvature R. Their definitions can be generaized to higher dimensions in an obvious way. Let R j i = g ik R kj, then the scaar curvature is R=R i i. Let us aso introduce the foowing object: 4.7 In the commutative case, p However, note that p :=g p, i E i,ẽ = g ik R kpi. 4.8 coincides with R p, but it is no onger true in the present setting. = g ik R ki = g ik R ki = R. 4.9 By first contracting the indices j and in the second Bianchi identity, then raising the index k to i by mutipying the resuting identity by g ik from the eft and summing over i, we obtain the identity 0= p R i R p i + g i, p E i,ẽ + g, p i E i,ẽ p + g i, E i, pẽ + g p, i E i, Ẽ + g p, i E i,ẽ + g, p E i, iẽ. Let us denote the sum of the ast two terms on the right-hand side by p. Then p = g ik r R kp ri r i g rk R kpi. In the commutative case, p vanishes identicay for a p. However, in the noncommutative setting, there is no reason to expect this to happen. Let us now define i R p;i = i R i i p pr R r i + ir i R r p, p; Then the second Bianchi identity impies = p r rp + r p r p i R p;i i + p;i p R = The above discussions suggest that Einstein s equation no onger takes its usua form in the noncommutative setting, but we have not been abe to formuate a basic principe which enabes us to derive a noncommutative anaog of Einstein s equation. However, formuas 4.11 and 4.9 seem to suggest that the foowing is a reasonabe proposa for a noncommutative Einstein equation in the vacuum: R j i + j i j i R = We were informed by Madore that in other contexts of noncommutative genera reativity, it aso i appeared to be necessary to incude an object anaogous to j in the Einstein equation. V. GENERAL COORDINATE TRANSFORMATIONS We investigate the effect of genera coordinate transformations on noncommutative n-dimensiona surfaces. This requires us to consider noncommutative surfaces defined over A Downoaded 04 Nov 2008 to Redistribution subject to ASCE icense or copyright; see

18 Chaichian et a. J. Math. Phys. 49, endowed with star products more genera than the Moya product. This shoud be compared to Refs. 9, 10, 12, and 13, where the ony genera coordinate transformations aowed were those keeping the Moya product intact. For the sake of being concrete, we assume that the noncommutative surface has Eucidean signature. A. Gauge transformations Denote by G A the set of R h -inear maps :A A satisfying the foowing conditions: 1 =1, = exp i i mod h, 5.1 i where i are smooth functions on U. Then ceary we have the foowing resut. Lemma 5.1: The set G A forms a subgroup of the automorphism group of A as R h -modue. For any given G A, define an R h -inear map, Lemma 5.2: :A R h A A, f g f g:= 1 f g The map is associative, thus there exists the associative agebra A, over R h. Furthermore, : A, A, is an agebra isomorphism. Let = 1, then for any, G A 1 f 1 g = f g. 5.3 In this sense the definition of the new star products respects the group structure of G A. Proof: Because of the importance of this emma for ater discussions, we sketch a proof for it here, even though one can easiy deduce a proof from Ref. 23. For f,g,h A, we have f g h = 1 f g h = 1 f g h = 1 f g h = f g h, which proves the associativity of the new star product. As is an R h -modue isomorphism by definition, we ony need to show that it preserves mutipications in order to estabish the isomorphism between the agebras. Now f g = f g. This proves part 1. Part 2 can be proven by unraveing the eft-hand side of 5.3. Adopting the terminoogy of Drinfed from the context of quantum groups, we ca an automorphism G A a gauge transformation and ca G A the gauge group. The star product wi be said to be gauge equivaent to the Moya product 4.1. However, note that our notion of gauge transformations is sighty more genera than that in deformation theory, 23 where the ony type of gauge transformations aowed is of the specia form, = id + h 1 + h 2 2 +, with i being R-inear maps on the space of smooth functions on U such that i 1 =0 for a i. Such gauge transformations form a subgroup of G A. Remark 5.3: The prime aim of the deformation theory 23 is to cassify the gauge equivaence casses of deformations in this restricted sense but for arbitrary associative agebras. The semina paper 4 of Kontsevich provided an expicit formua for a star product from each gauge equivaence cass of deformations of the agebra of functions on a Poisson manifod. Remark 5.4: Genera star products gauge equivaent to the Moya product were evauated Downoaded 04 Nov 2008 to Redistribution subject to ASCE icense or copyright; see

19 Riemannian geometry of noncommutative surfaces J. Math. Phys. 49, expicity up to the third order in h in Ref. 28. In Ref. 29, position-dependent star products were aso investigated and the utravioet divergences of a quantum 4 theory on four-dimensiona spaces with such products were anayzed. Given an eement in the group G A, we denote u i := 1 t i, i =1,2,...,n, and refer to t u as a genera coordinate transformation of U. Define R h -inear operators on A by i = 1 i. Lemma 5.5: The operators i have the foowing properties: 5.4 and aso satisfy the Leibniz rue i j j i =0, i 1 t j = i j, i f g = i f g + f i g, f,g A. Proof: The proof is easy but very iuminating. We have i j j i = 1 i j j i =0. Aso, i 1 t j = 1 i t j = i j, since maps a constant function to itsef. To prove the Leibniz rue, we note that i f g = 1 i f g = 1 i f g + 1 f i g = 1 i f g + 1 f i g = i f g + f i g. This competes the proof of the emma. The Leibniz rue pays a crucia roe in constructing noncommutative surfaces over A,. B. Reparametrizations of noncommutative surfaces The construction of noncommutative surfaces over A, works equay we over A, for any G A. Regard A m R h A m as a two-sided A, -modue and define the new dot product :A m R h A m A by A B=a i b i for any A= a 1,a 2,...,a m and B= b 1,b 2,...,b m in A m. It is obviousy a map of two-sided A, -modues. For an eement X = X 1,X 2, in the free two-sided A, -modue A m, we define...,x m E i = i X, g ij = E i E j, where acts on A m in a componentwise way. As in Sec. II, et g = g ij i,j=1,...,n. We sha say that X is an n-dimensiona noncommutative surface with metric g if g mod h is invertibe. In this case, g has an inverse g ij. Downoaded 04 Nov 2008 to Redistribution subject to ASCE icense or copyright; see

20 Chaichian et a. J. Math. Phys. 49, The eft tangent bunde TX and right tangent bunde T X of X are now, respectivey, the eft and right A, -modues generated by E i, i=1,2,...,n. The metric g eads to a two-sided A, -modue map, g:tx R h T X A, Z W Z W, which is the restriction of the dot product to TX R h T X. By using this map, we can decompose A m into A m = TX TX, as eft A, modue, A m = T X T X, as right A, modue, where TX is orthogona to T X and T X is orthogona to TX with respect to the map induced by the metric. As in Definition 2.5, the operators i :TX TX, i :T X T X are defined to be the compositions of i with the projections of A m onto the eft and right tangent bundes, respectivey. Thus, for any Z TX and W T X, g i Z,W = i Z W, g Z, i W = Z i W. 5.5 By using the Leibniz rue for i, we can show that the anaogous equations of 2.8 are satisfied by i and i, namey, for a Z TX, W T X, and f A, i f Z = i f Z + f i Z, i W f = W i f + i W f. Furthermore, the operators are metric compatibe, 5.6 i g Z,W = g i Z,W + g Z, i W, Z TX,W T X. Thus the two sets i and i define connections on the eft and right tangent bundes, respectivey. The Christoffe symbos k ij and k ij in the present context are aso defined in the same way as before, Then we have i E j = k ij E k, k ij = i E j E g k, i E j = E k k ij. k ij = g k E i E j. These formuas are of the same form as those in Eq By using 5.6, we can show that the maps i, j :TX TX and i, j :T X T X are eft and right A, -modue homomorphisms, respectivey. Namey, for a Z TX, W T X, and f A, i, j f Z = f i, j Z, Downoaded 04 Nov 2008 to Redistribution subject to ASCE icense or copyright; see

21 Riemannian geometry of noncommutative surfaces J. Math. Phys. 49, i, j W f = i, j W f. Thus we can define the curvatures R ijk, R ijk as before by i, j E k = R kij E, i, j E k = E R kij, and aso construct their reatives such as R ijk and R ijk. Then R ijk and R ijk are given by the formuas of Lemma 2.12 with repaced by, i by i, by, and by. Aso, Proposition 2.13 is sti vaid in the present case, and R ijk and R ijk satisfy the Bianchi identities see Theorem 4.3. Now we examine properties of noncommutative surfaces under genera coordinate transformations. Let X= X 1,X 2,...,X m be an eement of A m. We assume that X gives rise to a noncommutative surface over A,. Then we have the foowing reated noncommutative surfaces: Xˆ = X 1, X 2,..., X m, over A,, X = X 1,X 2,...,X m, over A,, associated with X. We ca X over A, the reparametrization of the noncommutative surface X over A, in terms of u= 1 t. Remark 5.6: Note that A m is regarded as an A, -modue when we study the noncommutative surface X and regarded as an A, -modue when we study X. Thus even though X and X are the same eement in A m, they have quite different meanings when the modue structures of A m are taken into account. Denote by ĝ the metric, by ˆ k ij and ˆ k ij the Christoffe symbos, and by Rˆ ijk and R ˆ ijk the Riemannian curvatures of Xˆ. They can be computed by using n-dimensiona generaizations of the reevant formuas derived in Sec. II. The metric, curvature, and other reated objects of the noncommutative surface X over A, are given in the ast subsection. Theorem 5.7: There exist the foowing reations: g ij = 1 ĝ ij, g ij = 1 ĝ ij, k ij = 1 ˆ k ij, ij k = 1 ˆ k, ij R ijk = 1 Rˆ ijk, R ijk = 1 R ˆ Remark 5.8: We sha see in the next section that this theorem eads to the standard transformation rues for the metric, connection, and curvature tensors in the commutative setting when we take the imit h 0. Proof of Theorem 5.7: Consider the first reation. Since 1 is an agebraic isomorphism from A, to A,, we have ijk. 1 ĝ ij = 1 i Xˆ j Xˆ = 1 i Xˆ 1 j Xˆ. Using 1 i Xˆ = i X, we obtain 1 Ê i = E i, i. Thus Downoaded 04 Nov 2008 to Redistribution subject to ASCE icense or copyright; see

22 Chaichian et a. J. Math. Phys. 49, ĝ ij = E i E j = g ij. Since maps 1 to itsef, it foows that 1 ĝ ij = g ij. Now 1 ˆ k ij = 1 i Ê j Ê ĝ k = 1 i Ê j 1 Ê 1 ĝ k = i E j E g k = k ij. The other reations can aso be proven simiary by using the fact that 1 is an agebraic isomorphism. We omit the detais. It is usefu to observe how the covariant derivatives transform under genera coordinate transformations. We have i E j = i E j + k ij E k = 1 i Ê j + 1 ˆ k ij Ê k = 1 ˆ iê j, where ˆ i is the covariant derivative in terms of the Christoffe symbos ˆ k ij. Remark 5.9: The gauge transformation that procures the genera coordinate transformation aso changes the agebra A, to A,, thus inducing a map between noncommutative surfaces defined over gauge equivaent noncommutative associative agebras. This is very different from what happens in the commutative case, but appears to be necessary in the noncommutative setting. Remark 5.10: Athough the concept of covariance under the gauge transformation is transparent and the considerations above show that our construction of noncommutative surfaces is indeed covariant under such transformations, it appears that the concept of invariance becomes more subte. In the cassica case, a scaar is a function on a manifod, which takes a vaue at each point of the manifod. Invariance means that when we evauate the function at the same point on the manifod, we get the same vaue a rea or compex number regardess of the coordinate system which we use for the cacuation. In the noncommutative case, eements of A are not numbers. When a genera coordinate transformation is performed, the agebraic structure of A changes. It becomes rather uncear how to compare eements in two different agebras. C. Comparison with cassica case One obvious question is the cassica anaogs of the differentia operators i and the gauge transformations in G A which bring about the genera coordinate transformations. We address this question beow. Moray, one shoud regard G A as the diffeomorphism group of the surface and i as differentiation with respect to the new coordinate u i := 1 t i. Consider an eement G A. In the cassica imit that is, h =0, obviousy reduces to an eement exp in the diffeomorphism group of U, where v= i t i is a smooth tangent vector fied on U cassicay. Then for any two smooth functions a t and b t, the Leibniz rue v ab =v a b+av b impies that exp v ab = exp v a exp v b at the cassica eve. By regarding a smooth function a t as a power series in t or t transated by some constants we then easiy see that Now for a f t A, exp v a t = a exp v t at the cassica eve. 5.7 i 1 f t = 1 f u i f t = u i mod h = 1 f t u i mod h, where we have used 5.7. Repacing f t by f t in the above computations we arrive at Using this resut, we obtain i f t = f t u i mod h. Downoaded 04 Nov 2008 to Redistribution subject to ASCE icense or copyright; see